Nonexistence of Anomalous Commutators in Spinor Electrodynamics

An attempt is made to infer the existence of so-called anomalous terms in the divergence of the axial-vector current from an action-principle approach. However, the presence of the spatial vector $\epsilon$ which is used to write the axial-vector current as the limit of a gauge-invariant nonlocal operator is found to preclude the possibility of successfully carrying out such a program. It is demonstrated that this failure is only one of several field-theoretical paradoxes which arise from the use of the $\epsilon$ limiting procedure. One is thereby led to the conclusion that the definition of current operators in terms of a limit can give consistent results only in a theory which is finite or one which is made finite by a regularization technique. The effect of such considerations is shown to imply that the Schwinger-Adler result for the axial-vector divergence is to be taken as a relation between matrix elements rather than one between field operators. Of more practical concern is the fact that this serves to resolve a discrepancy which has existed between several calculations of the commutators of ${j_5}^0$ with the electric charge density.